Terry was cleared of a racially aggravated public order offence at Westminster Magistrates' Court on Friday, after it had been alleged that he had used a racist obscenity about Queens Park Rangers defender Anton Ferdinand, brother of Rio.
Cole, 31, had told the trial that Terry, his Stamford Bridge captain and friend, was not racist.
Rio Ferdinand on Saturday responded to a tweet in which Cole was described as a 'choc ice', replying: "I hear you fella! Choc ice is classic! hahahahahahha!!"
Users began discussing the comments and reacting to them and Ferdinand wrote: "And if I want to laugh at something someone tweets....I will! Hahahahaha! Now stop getting ya knickers in a twist!"
Cole later issued a statement through his lawyers to stress that he did not want to take the matter any further.
The statement said: "Ashley Cole has been made aware of the discussion following comments appearing on Twitter and wishes to make it clear that he and Rio Ferdinand are good friends and Ashley has no intention of making any sort of complaint.
"Ashley appreciates that Tweeting is so quick it often results in off-hand and stray comments."
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Rio, who unlike Terry was left out of the England squad that went to Euro 2012, later referenced the Blues centre-half's legal defence by tweeting he was being sarcastic.
The 'choc ice' response was apparently deleted.
The Manchester United defender has also insisted he did not aim a racist taunt at Cole.
He tweeted: "What I said yesterday is not a racist term. Its a type of slang/term used by many for someone who is being fake. So there."